Mathematical Problems in Engineering

Volume 2016, Article ID 1593849, 13 pages

http://dx.doi.org/10.1155/2016/1593849

## Simulation of Partial and Supercavitating Flows around Axisymmetric and Quasi-3D Bodies by Boundary Element Method Using Simple and Reentrant Jet Models at the Closure Zone of Cavity

Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran

Received 3 February 2016; Accepted 10 April 2016

Academic Editor: Song Cen

Copyright © 2016 M. Nouroozi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A fixed-length Boundary Element Method (BEM) is used to investigate the super- and partial cavitating flows around various axisymmetric bodies using simple and reentrant jet models at the closure zone of cavity. Also, a simple algorithm is proposed to model the quasi-3D cavitating flows over elliptical-head bodies using the axisymmetric method. Cavity and reentrant jet lengths are the inputs of the problem and the cavity shape and cavitation number are some of the outputs of this simulation. A numerical modeling based on Navier-Stokes equations using commercial CFD code (Fluent) is performed to evaluate the BEM results (in 2D and 3D cases). The cavitation properties approximated by the present research study (especially with the reentrant jet model) are very close to the results of other experimental and numerical solutions. The need for a very short time (only a few minutes) to reach the desirable convergence and relatively good accuracy are the main advantages of this method.

#### 1. Introduction

The cavitating flows around various bodies have been the subject of extensive theoretical, numerical, and experimental research in recent decades. When the pressure of liquid becomes equal to or less than its saturated vapor pressure in fixed temperature conditions, the balance of fluid intermolecular forces is disrupted and subsequently a tensile stress occurs. The tensile stress leads to the development of vapor cavities in the liquid. Formation and growth of cavity usually occur when the liquid is subjected to sudden changes of pressure. This event can happen in many fluid systems like pumps, nozzles, turbine blades, and hydrofoils. In these systems, cavitation is known for its destructive effects like noise production, corrosion, and reduction of efficiency; therefore, researchers and engineers attempt to minimize these destructive effects. On the other hand, in some applications, cavitation can be used as a beneficial phenomenon. For example, in high-speed submerged vehicles, cavitation is desired since it leads to a significant reduction of drag force. This advantage of cavitation phenomenon is exploited to increase velocity and efficiency [1].

It has been observed that the main fluid flow passing over the cavity surface tends to return into the cavity from the end of it. This flow is called “reentrant jet.” Major reason for developing reentrant jet is a tendency of the fluid to move from the higher pressure zone (on the cavity surface) to the lower pressure zone (inside the cavity). Reentrant jet is developed under specific conditions. These conditions include (a) a high reverse pressure gradient at the closure of the cavity and (b) a considerable thickness of the cavity. Cavities of extremely large length or very small length do not satisfy (a) and (b) conditions, respectively. Therefore, small- and large-length cavities do not have a considerable reentrant jet, and only medium-length cavities are accompanied by reentrant jet [2]. Due to the various flow directions and phases at the end of the cavity, simulation of this zone is very complicated. In contrast to Navier-Stokes equations which do not require the end of cavity to be simulated, this zone should be modeled in BEM. Several models have been suggested for the cavity closure. In the present study, two models have been used for this critical zone of the cavity: simple closure model and reentrant jet model. In simple closure model, the cavity is closed on the body surface and consequently a stagnation point is formed at the end of the cavity. In the reentrant jet model, the flow over the cavity surface changes its direction into the cavity. The reentrant jet velocity is typically assumed to be of the order of the velocity of the flow over cavity surface. Reentrant jet closure model closely fits the behavior of unsteady cavitating flows.

BEM has potential flow as a basic assumption. As a result, one can use BEM for cavitation analysis only when cavitating flow is proven to be potential. Experimental studies of Labertaux and Ceccio show that the flow around cavity is reasonably approximately a potential one [3]. Therefore, potential theories (such as BEM) may be utilized for the simulation of cavity around bodies. BEM is based on “Green theory.” Green theory states that every incompressible and irrotational flow can be simulated by source, dipole, or vortex distribution on its boundary surfaces [4]. In the present work, dipole and source rings have been distributed on the body/cavity boundaries to simulate cavitating flows over axisymmetric geometries.

The first studies for cavitation were carried out by Efros [5] and Gilbarg and Serrin [6] using 2D analytical theory of free streamlines. After that, modeling of cavitation over hydrofoils using the linear theory of flow was introduced. Tulin [7, 8] and Guerst [9] developed this method. After Hess and Smith [10], who calculated potential flow around arbitrary bodies, the application of this method rapidly increased. For the first time, Uhlman [11] used a nonlinear BEM based on velocity to solve the 2D partial cavitating flows around hydrofoil using a vortex distribution over the flow boundaries. They proceeded to use the same method to solve the 2D supercavitating flow around hydrofoil [12]. Kinnas and Fine [13] offered another nonlinear BEM based on potential to solve partial cavitating flow on 2D hydrofoil. They started solving this problem by distributing source and dipole on the boundaries of the flow using Green’s third identity. The convergence and accuracy of potential-based BEM were better than those of the velocity-based BEM. After that, Birkhoff and Zarantonello [14] and Gilbarg [15] investigated formation of a reentrant jet of the cavity in symmetric flow against a vertical flat plate for the first time; other researchers such as Pellone and Rowe [16], Fine and Kinnas [17], and Vaz [18] employed the reentrant jet model for the cavity termination. Also, Uhlman [19], Nouri et al. [20], and Rashidi et al. [21] used potential-based BEM for cavitating flows over axisymmetric bodies. Uhlman et al. used this method for supercavitating flows using reentrant jet model and Nouri et al. solved these flows using simple closure model. Pasandideh-fard et al. applied the method together with simple closure model for partial and supercavitating flows.

In this work, a BEM analysis of partial and supercavitating flows around axisymmetric bodies is presented and two models of termination of the cavity (simple model and reentrant jet) are compared. To simulate these flows, body, cavity, and jet boundaries are approximated by elements and source and dipole rings are distributed on them, based on Green’s third identity. In addition, modeling of cavitating flows is performed with zero angle of attack over quasi-3D bodies (cylinders with elliptical head) using present axisymmetric code. To evaluate the results, 2D and 3D numerical commercial CFD codes (Fluent) are executed. The present work, in a novel manner, employs the reentrant jet model at the end of partial cavity over different axisymmetric geometries. Besides, the main innovation of this research is the exploitation of the rapid convergence capability of axisymmetric BEM in the modeling of cavitating flows over elliptical-head cylinders (quasi-3D flow). This algorithm had not been reported in the literature. It is noteworthy that the present quasi-3D analysis yields relatively appropriate results at minimal computational and time cost (only in a few minutes), whereas the other numerical methods involve greater complication and much higher cost (they take at least hours).

#### 2. Mathematical Equations

The cylindrical coordinate system is the best coordinate system for axisymmetric flows. Integral expression of Green’s third identity in this coordinate system is as follows: where is the normal vector directed outward from the solid-body surface and the cavity interface, is arc length along a meridian, and are the components of the axisymmetric coordinate system, and is “disturbance” potential on the solved surfaces. In fact, disturbance potential, , is the sum of all the potential flow elements, except free stream potential, in one point [11]. Thus, the total () and disturbance () potentials are related by where is the free stream velocity that flows on the body surface. Considering to be equal to unit, (2) becomes dimensionless:Since the bodies are axisymmetric, distributed potential elements on the body/cavity surfaces should be rings to be appropriate for the cylindrical coordinate system. and are the potential functions related to the sources and dipoles distributed around a ring, respectively [3]. Source and dipole rings used in Green theory can be obtained by integrating these potential elements around the axis [19]. The potential function of a source ring () with radius at point , when the ring center is located at and for the unit surface of , is defined as follows:And the potential function of a dipole ring () that is the normal derivative of the source ring is defined as follows: Cylindrical coordinate components used in (4) and (5) are shown in Figure 1.